The L¹ Ehrenpreis Conjecture

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Let S be the algebraic universal cover of a closed surface of genus >1, T(S) its Teichmuller space, M(S) the group of mapping classes stabilizing a fixed leaf l. The L¹ Ehrenpreis conjecture asserts that M(S) on T(S) with dense orbits with respect to the L¹ topology (the topology induced by the L¹ norm on quadratic differentials). We give a proof of this weaker form of the Ehrenpreis conjecture.